
where we denote the region sandwiched between \({A_i}\)  and \({A_j}\)  as \(\tilde{A} = {A_{\left( {i + 1} \right) \cdots \left( {j - 1} \right)}}\)  . See paper [1]}, and it is not difficult to find that this expression is equivalent to the so-called PEE proposal proposed in [2]}, [3]}. Moreover, in fact, PEE is referred to as conditional mutual information (CMI) in quantum information theory (up to a conventional 1/2 factor) [4]}, [5]}, and (REF ) is just the definition of CMI. This is for each component flow flux (CFF) to exactly match the partial entanglement entropy (PEE) in the boundary quantum system one by one, so as to reflect the entanglement structure information of the boundary quantum system correctly and reasonably. Using the convex dualization technique in the convex optimization theory and the bulk-cell decomposition method proposed in [6]}(see also [7]}), we can prove that a locking thread configuration can lock a specified set of non-intersecting boundary regions simultaneously while subject to the extra constraints (REF ) at the same time always exists. See [8]} for details.
